论文标题
关于对称和遗传学等级距离代码
On symmetric and Hermitian rank distance codes
论文作者
论文摘要
令$ \ cal m $表示集合$ {\ cal s} _ {n，q} $ of $ n \ times n $ n $对称矩阵，带有$ {\ rm gf}（q）$的条目或set $ {\ cal h} _ {\ cal h} _ { gf}（q^2）$。然后，配备了等级距离$ d_r $的$ \ cal m $是一个度量空间。我们研究了$（{\ cal m}，d_r）$中的$ d $ - 码，并构建$ d $ - 代码，其大小大于相应的加法界。在赫米尔人的情况下，我们表明存在$ \ cal m $，$ n $偶数和$ n/2 $奇数的$ n $ code的存在（3q^{n} -q^{n} -q^{n/2} \ right）/2 $，以及$ 2 $ $ q^6+q^6+q^6+q（q-q-n $ q（q^q^^q^4+q^4+q^4+q^4+q^4+q^4+q^4+q^4+q^4+q^4+q^4+q^4+q^4+q^4+Q^4+q^4+q.）在对称的情况下，如果$ n $是奇数，或者$ n $和$ q $均为奇数，那么我们提供更好的上限，以$ 2 $代码的大小。在$ n = 3 $和$ q> 2 $的情况下，展示了$ 2 $的$ q^4+q^3+1 $的代码。这提供了第一个由$ 2 $ 2 $的对称矩阵的无限家族，其大小大于最大的添加剂$ 2 $代码。
Let $\cal M$ denote the set ${\cal S}_{n, q}$ of $n \times n$ symmetric matrices with entries in ${\rm GF}(q)$ or the set ${\cal H}_{n, q^2}$ of $n \times n$ Hermitian matrices whose elements are in ${\rm GF}(q^2)$. Then $\cal M$ equipped with the rank distance $d_r$ is a metric space. We investigate $d$-codes in $({\cal M}, d_r)$ and construct $d$-codes whose sizes are larger than the corresponding additive bounds. In the Hermitian case, we show the existence of an $n$-code of $\cal M$, $n$ even and $n/2$ odd, of size $\left(3q^{n}-q^{n/2}\right)/2$, and of a $2$-code of size $q^6+ q(q-1)(q^4+q^2+1)/2$, for $n = 3$. In the symmetric case, if $n$ is odd or if $n$ and $q$ are both even, we provide better upper bound on the size of a $2$-code. In the case when $n = 3$ and $q>2$, a $2$-code of size $q^4+q^3+1$ is exhibited. This provides the first infinite family of $2$-codes of symmetric matrices whose size is larger than the largest possible additive $2$-code.
